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It obviously does, otherwise there wouldn't be anything displayed. It's probably a GMA chip, which while very weak for gaming, is fine for average desktop stuff. It's also supported well by Linux. As long as the Atom has hyper-threading (or is dual-core), it should run fine even on shitty GMA integrated graphics.

Enable the hiding option. Just put your most-used programs at the top, and access them with start+number keys so you don't have to use the mouse. Bring up the Dash with Start. Make your panel icons smaller.

As long as you have a decently supported 3D acceleration (Intel graphics should work fine, which is what you probably have), Unity should work. If it's a single-core processor, then it'll be slow. Check if your Atom has Hyper-Threading support. Or just open System Monitor and check if there are two lines in the CPU graph. If there are, you're fine in terms of resources.

Just download the ISO and make a LiveUSB. Boot it up, see how well it runs. Best way to see is to try for yourself. Keep in mind that the loading speeds of a LiveUSB are slower than installing on the internal HDD.

If it's too slow for you, you can install Xubuntu. Fairly configurable, and you can install something like Gnome DO for a Dash replacement.

No, it still uses the same amount of RAM. The difference is that 32-bit programs cannot use more than 2GB of RAM, while 64-bit programs can use a LOT more than that. Doesn't mean that they will use that much.

If you're worried about memory, then install something like Xubuntu (64-bit, if your CPU supports it). If you're happy with Unity's memory usage, though, go ahead and use the regular Ubuntu.

TL;DR: If your CPU supports 64-bit operating systems, then use the 64-bit version. It doesn't use more or less memory than 32-bit.

I have a an almost equally shitty lappy, and am really loving mint 13, uses precise repos, but has mate pre-installed. You could just remove whatever DE and install mate, but last time I tried that on ubuntu it messed up plymouth and I would have to open a new terminal to log in and start X.